# The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper
Bounds

Research paper by **Kenley Jung**

Indexed on: **15 Feb '16**Published on: **15 Feb '16**Published in: **Mathematics - Operator Algebras**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

Using an analogy with the rank theorem in differential geometry, it is shown
that for a finite $n$-tuple $X$ in a tracial von Neumann algebra and any finite
$m$-tuple $F$ of $*$-polynomials in $n$ noncommuting indeterminates,
\begin{eqnarray*} \delta_0(X) & \leq & \text{Nullity}(D^sF(X)) +
\delta_0(F(X):X) \end{eqnarray*} where $\delta_0$ is the (modified) microstates
free entropy dimension and $D^sF(X)$ is a kind of derivative of $F$ evaluated
at $X$. When $F(X) =0$ and $|D^sF(X)|$ has nonzero Fuglede-Kadison-L\"uck
determinant, then $X$ is $\alpha$-bounded in the sense of \cite{j3} where
$\alpha = \text{Nullity}(D^sF(X))$. Using Linnell's $L^2$ integral domain
results in \cite{l} as well as Elek and Szab\'o's work on L\"uck's determinant
conjecture for sofic groups in \cite{es} the following result is proven.
Suppose $\Gamma$ is a sofic, left-orderable, discrete group with 2 generators
and $\Gamma \neq \{0\}$. The following conditions are equivalent:
(1) $\Gamma \not\simeq \mathbb F_2$.
(2) $L(\Gamma) \not\simeq L(\mathbb F_2)$.
(3) $L(\Gamma)$ is strongly $1$-bounded.
(4) $\delta_0(X) = 1$ for any finite set of generators $X$ for $L(\Gamma)$.
From Brodski\u{i} and Howie's results on local indicability (\cite{b},
\cite{h}), it follows that a sofic, torsion-free, one-relator group von Neumann
algebra on two generators with nontrivial relator is strongly $1$-bounded. It
also follows from the residual solvability of the positive one relator groups
(\cite{baum}) that a one-relator group von Neumann algebra on two generators
whose relator is a nontrivial, positive, non-proper word in the generators is
strongly $1$-bounded.